Euler's Formula - Numberphile

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NUMBERPHILE

Videos by Brady Haran

[elgalas]

We need a video explaining why the polar representation of a complex number includes "e". It's a wonderful bit by Euler.

[AtricosHU]

This is not actually how the formula is derived. You've just *used* Euler's formula to *derive* Euler's formula. Which is great, at least you found no inconsistencies in maths, but it doesn't explain where the z = r * e^(i * theta) comes from. Why is this equality true? Why can you represent z using the exponential of i times the angle? This *comes* from Euler's formula in the first place, which can be proven looking at the Taylor series of sin, cos, and exp, respectively.

[avhuf]

it's more intuitive without the zero, then you can see that exp(i*pi) is actually a rotation over pi starting from 1, and thus landing on -1.

[harper]

personal favourite part of euler's definitions of trigonometric functions is how neatly they relate standard functions to hyperbolic functions, whereas only considering the reals they look nothing like each other - neat example is that sin(z) rotated pi/2 around the plane is the same graph as i*sinh(z)

[johnny_eth]

The coolest feature of the polar representation is that it makes calculating roots or powers super super easy. Just divide or multiply the angle, and don't forget the 2πn term.

[markcrites]

I didn't hear him explicitly state this, but for all of this to work theta must be in radians. Later on in the video he starts mixing degrees and radians. This is all common sense to a mathematician but it may not be to a student early in their math career.

[mestar]

Magic moment happens at 1.55. He's explaining basics of complex numbers, and then he just right into using "e to the power of i something", which makes no sense. You need to justify using that magic at that point.

[andmefikri]

I want to say thank you! Thank you from the bottom of my heart. I am a first year university student with math major, and so far, my experience have been discouraging me away from math. I kept thinking that I made a mistake, that I made the wrong choice, and that mathematics isn't for me. This video has reminded me of my love and passion of math. I took the bargain, math isn't easy, but it rewards your curiosity and perseverance.

[Bodyknock]

2:08 Tom says "and the polar form is z = reᶦᵒ ..." and then goes on to say how that implies Euler's Formula. But in order to use z = reᶦᵒ as the polar form you're already implicitly assuming that eᶦᵒ = cosθ + isinθ so it's a bit of circular reasoning. It's not at all obvious without having already proven Euler's formula that if z = reᶦˣ that x is actually the angle θ in the polar form r(cosθ + isinθ).

I think the proof he was actually intending to use starts off by saying "Assume z = eᶦˣ . (We're not making any assumption other than that it's a complex number, we know nothing about x or its relation to the polar form so far. The only thing we know is that it's a complex number.) Let z = r(cosθ + isinθ) be its polar form. So eᶦˣ = r(cosθ + isinθ) and r and θ are functions of x. Now we want to find a closed form for x in terms of r and θ..." and go from there.

[jmcsquared]

1:51 This was a bit of a cheat, because that's not really polar form. That every complex number can be written as z=r⋅e^i𝜃 assumes Euler's formula, as well as begs the question of what we could possibly mean by e^i𝜃 in the first place (i.e., raising something to an imaginary power). The actual polar form of a complex number is z=r⋅(cos(𝜃)+isin(𝜃)) which is easily shown by your geometric argument with cosines and sines in triangles. The theorem behind Euler's formula is that the trigonometric part in parentheses cos(𝜃)+isin(𝜃) is equal to the exponential function e^i𝜃 once it's been suitably defined. Of course, you show that using the power series expansions later on, but the presentation is quite out of order, which could confuse viewers.

[bjornmu]

This is all cool, but you don't explain how e "magically" enters the formula in the first place. How come re^(i phi) equals x +iy?

[joaobaptista]

How you represented complex numbers in polar form using exponential without allready knowing Euler's formula? It looks like circular reasoning to me.

[Israel..]

Euler's work is so much fun.
A, B, C, ..., N, ... for any sequence whatever, index notation is rare which forces a sort of conceptual elegance in the arguments.

[genessab]

Wait you can’t start your proof with the polar form of complex numbers, does that not presuppose Euler’s formula?

[jessehammer]

Wow, there’s never been a Numberphile about this? Color me surprised (but excited too!).

[frankheyder]

That leaves the question: Is Numberphile an infinite series? I surly hope so.

[happyestus]

I love it when Tom's doing the explanation, he's always so enthusiastic! Very contagious energy

[calmkat]

This whole Euler Formula to Trigonometry thing is also my favorite thing in math! The connection between a Taylor Series in the specific case of e^x being the same as cos(x) * isin(x), and how that solves so many problems in trig, immediately solved so many hard-to-grasp concepts in calculus and physics that plagued me throughout all of high school.

[ijuhat]

I prefer to think of Euler's formula as the *definition* of the cosine and sine functions. That is, you start by defining complex numbers, then define the exponential function (on the complex domain) via its power series, and then define cosine and sine to be the real and imaginary parts of this function. The really cool part is that you can then define pi analytically, by defining pi/2 to be the smallest positive root of cosine. Everything else falls out nicely after that. The way things are presented in this video suffers from circular reasoning in my view.

[vspirit]

11:05 i never thought it would bother me so much that -1 is on the right side